What if you had a triangle in which one leg was one unit shorter than the other and the hypotenuse was 5 units? How could you find the length of the two legs? After completing this Concept, you'll be able to solve real-world applications like this one that involve factoring polynomial equations.

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Guidance

Now that we know most of the factoring strategies for quadratic polynomials, we can apply these methods to solving real world problems.

Example A

One leg of a right triangle is 3 feet longer than the other leg. The hypotenuse is 15 feet. Find the dimensions of the triangle.

Solution

Let
the length of the short leg of the triangle; then the other leg will measure
.

Use the Pythagorean Theorem:
, where
and
are the lengths of the legs and
is the length of the hypotenuse. When we substitute the values from the diagram, we get
.

In order to solve this equation, we need to get the polynomial in standard form. We must first distribute, collect like terms and
rewrite
in the form “polynomial = 0.”

Factor
out the common monomial:

To factor the trinomial inside the parentheses, we need two numbers that multiply to -108 and add to 3. It would take a long time to go through all the options, so let’s start by trying some of the bigger factors:

We factor the expression as
.

Set each term equal to zero
and
solve:

It makes no sense to have a negative answer for the length of a side of the triangle, so the answer must be
. That means
the short leg is 9 feet and the long leg is 12 feet.

Check:
, so the answer checks.

Example B

The product of two positive numbers is 60. Find the two numbers if one number is 4 more than the other.

Solution

Let
one of the numbers; then
is the other number.

The product of these two numbers is 60, so we can write the equation
.

In order to solve we must write the polynomial in standard form. Distribute, collect like terms and
rewrite:

Factor
by finding two numbers that multiply to -60 and add to 4. List some numbers that multiply to -60:

The expression factors as
.

Set each term equal to zero
and
solve:

Since we are looking for positive numbers, the answer must be
.
One number is 6, and the other number is 10.

Check:
, so the answer checks.

Example C

A rectangle has sides of length
and
. What is
if the area of the rectangle is 48?

Solution

Make a sketch of this situation:

Using the formula Area = length
width, we have
.

In order to solve, we must write the polynomial in standard form. Distribute, collect like terms and
rewrite:

Factor
by finding two numbers that multiply to -63 and add to 2. List some numbers that multiply to -63:

The expression factors as
.

Set each term equal to zero
and
solve:

Since we are looking for positive numbers the answer must be
. So
the width is
and the length is
.

Vocabulary

Polynomials can be written in
expanded form
or in
factored form
. Expanded form means that you have sums and differences of different terms:

The
factored form
of a polynomial means it is written as a product of its factors.

Zero Product Property:
The only way a product is zero is if one or more of the terms are equal to zero:

We say that a polynomial is
factored completely
when we factor as much as we can and we are unable to factor any more.

Guided Practice

Consider the rectangle in Example C with sides of length
and
. What is
if the area of the rectangle is now 20?

Solution

Make a sketch of this situation:

Using the formula Area = length
width, we have
.

In order to solve, we must write the polynomial in standard form. Distribute, collect like terms and
rewrite:

Factor
by finding two numbers that multiply to -35 and add to 2. List some numbers that multiply to -35:

The expression factors as
.

Set each term equal to zero
and
solve:

Since we are looking for positive numbers the answer must be
. So
the width is
and the length is
.

Check:
, so the answer checks.

Practice

Solve the following application problems:

One leg of a right triangle is 1 feet longer than the other leg. The hypotenuse is 5. Find the dimensions of the right triangle.

One leg of a right triangle is 7 feet longer than the other leg. The hypotenuse is 13. Find the dimensions of the right triangle.

A rectangle has sides of
and
. What value of
gives an area of 108?

A rectangle has sides of
and
. What value of
gives an area of 120?

The product of two positive numbers is 120. Find the two numbers if one numbers is 7 more than the other.

A rectangle has a 50-foot diagonal. What are the dimensions of the rectangle if it is 34 feet longer than it is wide?

Two positive numbers have a sum of 8, and their product is equal to the larger number plus 10. What are the numbers?

Two positive numbers have a sum of 8, and their product is equal to the smaller number plus 10. What are the numbers?

Framing Warehouse offers a picture framing service. The cost for framing a picture is made up of two parts: glass costs $1 per square foot and the frame costs $2 per foot. If the frame has to be a square, what size picture can you get framed for $20?

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